Opuscula Math. 38, no. 5 (2018), 699-718
https://doi.org/10.7494/OpMath.2018.38.5.699

Opuscula Mathematica

On one condition of absolutely continuous spectrum for self-adjoint operators and its applications

Eduard Ianovich

Abstract. In this work the method of analyzing of the absolutely continuous spectrum for self-adjoint operators is considered. For the analysis it is used an approximation of a self-adjoint operator $$A$$ by a sequence of operators $$A_n$$ with absolutely continuous spectrum on a given interval $$[a,b]$$ which converges to $$A$$ in a strong sense on a dense set. The notion of equi-absolute continuity is also used. It was found a sufficient condition of absolute continuity of the operator $$A$$ spectrum on the finite interval $$[a,b]$$ and the condition for that the corresponding spectral density belongs to the class $$L_p[a,b]$$ ($$p\ge 1$$). The application of this method to Jacobi matrices is considered. As one of the results we obtain the following assertion: Under some mild assumptions, suppose that there exist a constant $$C\gt 0$$ and a positive function $$g(x)\in L_p[a,b]$$ ($$p\ge 1$$) such that for all $$n$$ sufficiently large and almost all $$x\in[a,b]$$ the estimate $$\frac{1}{g(x)}\le b_n(P_{n+1}^2(x)+P_{n}^2(x))\le C$$ holds, where $$P_n(x)$$ are 1st type polynomials associated with Jacobi matrix (in the sense of Akhiezer) and $$b_n$$ is a second diagonal sequence of Jacobi matrix. Then the spectrum of Jacobi matrix operator is purely absolutely continuous on $$[a,b]$$ and for the corresponding spectral density $$f(x)$$ we have $$f(x)\in L_p[a,b]$$.

Keywords: self-adjoint operators, absolutely continuous spectrum, equi-absolute continuity, spectral density, Jacobi matrices.

Mathematics Subject Classification: 47A10, 47A58.

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